U.S. patent application number 13/635857 was filed with the patent office on 2013-06-27 for apparatus and method of vibration control.
This patent application is currently assigned to UNIVERSITY OF SOUTHAMPTON. The applicant listed for this patent is Stephen John Elliott, Paolo Gardonio, Michele Zilletti. Invention is credited to Stephen John Elliott, Paolo Gardonio, Michele Zilletti.
Application Number | 20130166077 13/635857 |
Document ID | / |
Family ID | 42228008 |
Filed Date | 2013-06-27 |
United States Patent
Application |
20130166077 |
Kind Code |
A1 |
Elliott; Stephen John ; et
al. |
June 27, 2013 |
APPARATUS AND METHOD OF VIBRATION CONTROL
Abstract
Vibration control apparatus for controlling vibration of a
structure (6), the apparatus having an inertial actuator (1), a
velocity sensor (4) to measure the velocity of vibration of the
structure, and a controller (2) to provide a gain control signal to
the actuator. The controller is arranged to determine the gain
control signal using at least a measure of velocity from the
velocity sensor and a measure of force applied by the actuator to
the structure. The controller is further arranged to use the
measure of velocity and the measure of force applied to determine a
measure of power absorbed by the actuator, and to use the measure
of power to determine the gain control signal.
Inventors: |
Elliott; Stephen John;
(Hampshire, GB) ; Zilletti; Michele; (Hampshire,
GB) ; Gardonio; Paolo; (Portogruaro, IT) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Elliott; Stephen John
Zilletti; Michele
Gardonio; Paolo |
Hampshire
Hampshire
Portogruaro |
|
GB
GB
IT |
|
|
Assignee: |
UNIVERSITY OF SOUTHAMPTON
Southampton, Hampshire
GB
|
Family ID: |
42228008 |
Appl. No.: |
13/635857 |
Filed: |
March 18, 2011 |
PCT Filed: |
March 18, 2011 |
PCT NO: |
PCT/GB2011/050538 |
371 Date: |
March 7, 2013 |
Current U.S.
Class: |
700/280 |
Current CPC
Class: |
F16F 15/002 20130101;
G05D 19/02 20130101 |
Class at
Publication: |
700/280 |
International
Class: |
G05B 6/02 20060101
G05B006/02 |
Foreign Application Data
Date |
Code |
Application Number |
Mar 19, 2010 |
GB |
1004630.8 |
Claims
1. Vibration control apparatus for controlling vibration of a
structure, the apparatus comprising, an inertial actuator, a
velocity sensor to measure the velocity of vibration of the
structure, and a controller to provide a gain control signal to the
actuator, wherein, the controller arranged to determine the gain
control signal using at least a measure of velocity from the
velocity sensor and a measure of force applied by the actuator to
the structure, and wherein the controller arranged to use the
measure of velocity and the measure of force applied to determine a
measure of power absorbed by the actuator, and the controller
further arranged to use the measure of power to determine the gain
control signal.
2. Apparatus as claimed in claim 1, the controller arranged to
calculate the measure of power absorbed by determining the product
of the measure of velocity and the measure of force applied.
3. Apparatus as claimed in claim 1, the controller arranged to
determine the measure of force applied using the gain control
signal sent to the actuator.
4. Apparatus as claimed in any of claims 1 to 3 comprising a force
sensor to measure the force applied by the actuator to provide to
the controller a measure of the force applied.
5. Apparatus as claimed in claim 1 in which the velocity sensor
comprises an accelerometer.
6. Apparatus as claimed in claim 1, the velocity sensor arranged to
be attached to the structure and local to the inertial
actuator.
7. Apparatus as claimed in claim 1 which comprises a compensator to
reduce the apparent natural frequency of the actuator.
8. Apparatus as claimed in claim 7 in which the compensator
comprises a null to compensate for the natural frequency of the
actuator and a resonance of a frequency lower than the apparent
natural frequency.
9. Apparatus as claimed in claim 1 in which the controller has been
configured during an initial set-up procedure during which a
measured on-line response of the velocity sensor to the control
signal is used to suitably configure the controller.
10. Apparatus as claimed in claim 9 in which the controller has
been configured during the initial set-up procedure using an
actuator response and the response is deduced from the measured
on-line response of the velocity sensor.
11. Apparatus as claimed in claim 7 in which the compensator has
been configured during an initial set-up procedure using an
actuator response deduced from on-line measurements of the response
of the velocity sensor.
12. A controller for a vibration control apparatus, the controller
comprising a processor, the processor arranged to receive an input
indicative of a measure of velocity of vibration of a structure and
an input indicative of a measure of force applied to the structure
by an inertial actuator, and the processor arranged to provide a
gain control signal for the inertial actuator using at least the
measure of velocity and the measure of force applied, and wherein
the controller arranged to use the measure of velocity and the
measure of force applied to determine a measure of power absorbed
by the actuator, and the controller further arranged to use the
measure of power to determine the gain control signal.
13. A method of controlling vibration in a structure using an
inertial actuator, the method comprising, determining a measure of
velocity of vibration of the structure, determining a measure of
force applied by the actuator, using at least the measure of
velocity and the measure of force to determine a gain control
signal to the actuator, and using the measure of velocity and the
measure of force applied to determine a measure of power absorbed
by the actuator, and using the measure of power to determine the
gain control signal.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority from British Patent
Application GB 1004630.8, filed Mar. 19, 2010, and corresponding
International Patent Cooperation Treaty Application No.
PCT/GB2011/050538, filed Mar. 18, 2011, each fully incorporated
herein in their entirety.
TECHNICAL FIELD
[0002] The present invention relates generally to an apparatus and
a method of vibration control
BACKGROUND OF THE INVENTION
[0003] The active control of vibration on large structures requires
multiple actuators and sensors. The complexity of such a control
system scales linearly with the number of actuators and sensors if
these are arranged in collocated pairs and controlled using only
local, decentralised, feedback. Although the use of such a modular
approach to active control has several attractions, to provide good
performance they must be able to self-tune their feedback gain to
adapt to the environment they find themselves in.
[0004] There are a number of advantages to using multiple local
feedback loops to control the vibration in structures. These
include a complexity that only rises with the number of actuators,
a robustness to failure of individual loops and the possibility of
mass producing modular systems, including the actuator, sensor and
feedback loop.
[0005] One important issue with such an arrangement, however, is
how the feedback gains are set in the individual loops. The optimum
feedback gain is generally a compromise between performance and
stability, and its value changes for each loop on a particular
structure depending on its position on the structure, the type of
vibration and the state of all the other feedback loops. We have
realized that the feedback gain of each controller could be
adjusted, using only local parameters, to minimize the global
vibration of the structure, and this self-tuning would continue in
case there were any change in the conditions with time.
[0006] We seek to provide an improved apparatus and method of
vibration control.
SUMMARY OF THE INVENTION
[0007] According to a first aspect of the invention there is
provided a vibration control apparatus for controlling vibration of
a structure, the apparatus comprising, an inertial actuator, a
velocity sensor to measure the velocity of vibration of the
structure, and a controller to provide a gain control signal to the
actuator, wherein, the controller arranged to determine the gain
control signal using at least a measure of velocity from the
velocity sensor and a measure of force applied by the actuator to
the structure.
[0008] The controller may be arranged to use the measure of
velocity and the measure of force applied to determine a measure of
power absorbed by the actuator, and the controller further arranged
to use the measure of power to determine the gain control
signal.
[0009] The controller may be arranged to calculate the measure of
power absorbed by determining the product of the measure of
velocity and the measure of force applied.
[0010] The controller is preferably arranged to determine the
measure of force applied using the gain control signal sent to the
actuator.
[0011] The apparatus may comprise a force sensor to measure the
force applied by the actuator to provide to the controller a
measure of the force applied.
[0012] The velocity sensor may comprise an accelerometer.
[0013] The velocity sensor may be arranged to be attached to the
structure and local to the inertial actuator.
[0014] The apparatus may comprise a compensator to reduce the
apparent natural frequency of the actuator.
[0015] The compensator preferably comprises a null to compensate
for the natural frequency of the actuator and a resonance of a
frequency lower than the apparent natural frequency.
[0016] The controller is preferably such that it has been
configured during an initial set-up procedure during which a
measured on-line response of the velocity sensor to the control
signal is used to suitably configure the controller.
[0017] Preferably, the controller has been configured during the
initial set-up procedure using an actuator response and the
response is deduced from the measured on-line response of the
velocity sensor.
[0018] Preferably, the compensator has been configured during an
initial set-up procedure using an actuator response deduced from
on-line measurements of the response of the velocity sensor.
[0019] According to a second aspect of the invention there is
provided a controller for a vibration control apparatus, the
controller comprising a processor, the processor arranged to
receive an input indicative of a measure of velocity of vibration
of a structure and an input indicative of a measure of force
applied to the structure by an inertial actuator, and the processor
arranged to provide a gain control signal for the inertial actuator
using at least the measure of velocity and the measure of force
applied.
[0020] The controller preferably includes machine-readable
instructions to be executed by the processor.
[0021] According to a third aspect of the invention there is
provided a method of controlling vibration in a structure using an
inertial actuator, the method comprising, determining a measure of
velocity of vibration of the structure, determining a measure of
force applied by the actuator, using at least the measure of
velocity and the measure of force to determine a gain control
signal to the actuator.
[0022] In a preferred embodiment of the invention self-tuning of
local velocity feedback controllers is effected based on the
maximisation of their absorbed power, as estimated from the
measured velocity signal. For broadband excitations, maximisation
of the power absorbed, which requires only local measurements,
provides a good approximation to the minimisation of the overall
kinetic energy in a structure, corresponding to its global
response.
BRIEF DESCRIPTION OF THE DRAWINGS
[0023] Various embodiments of the invention will now be described,
by way of example only, with reference to the following drawings in
which:
[0024] FIG. 1 shows a power spectral density,
[0025] FIG. 2 shows a table,
[0026] FIGS. 3(a) and 3(b) show the frequency-averaged kinetic
energy distributions for different conditions,
[0027] FIG. 4 shows a self-tuning arrangement for direct velocity
feedback control with an ideal force actuator,
[0028] FIG. 5 shows the blocked frequency response of a single
degree of freedom model,
[0029] FIG. 6 shows the kinetic energy on a panel,
[0030] FIGS. 7(a) and 7(b) are plots of the frequency-averaged
kinetic energy of a panel and local absorbed power is plotted as a
function of feedback gain,
[0031] FIG. 8 shows an active vibration control apparatus with an
inertial actuator, and
[0032] FIG. 9 show plots of frequency averaged kinetic energy and
power absorbed by the controller.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0033] FIG. 1 shows the power spectral density (PSD) of the kinetic
energy on a panel of a structure, having the parameters listed in
the table shown in FIG. 2, for various values of the feedback gain,
.gamma., of a single feedback loop on the panel, in which the
measured velocity is fed back to a collocated force actuator. In
FIG. 1, the PSD of the panel's kinetic energy is shown with a local
velocity feedback loop driving an ideal force actuator with
feedback gains of .gamma.=0 (solid line), .gamma.=7 (dashed line),
.gamma.=25 (faint line) and .gamma.=10.sup.3 (dotted line). A modal
model of the panel is used, which is assumed to be excited by a
spatially random white noise signal with a bandwidth from 1 Hz to 1
kHz. When the feedback gain is zero, the original modes of the
panel can clearly be seen, and the low order modes are
progressively damped as the feedback gain is increased. Beyond a
certain gain, however, the feedback loop begins to pin the
structure instead of damping it, and a new set of relatively
undamped resonances begin to emerge. The local feedback loop with
this idealised actuator is acting as a skyhook damper with a
damping value determined by the feedback gain. The negative
feedback loop can thus only ever absorb mechanical power from the
structure, and so the feedback controller is unconditionally
stable.
[0034] If the frequency-averaged kinetic energy of the panel is
calculated for each condition, its variation with feedback gain,
normalised by the condition with no control, is shown in FIG. 3(a).
It initially decreases as the feedback gain is increased, before
increasing again as the controller begins to pin the panel. The
optimum feedback gain is approximately equal to the reciprocal of
the infinite panel's input mobility. FIG. 3(b) shows the frequency
averaged power absorbed by the feedback loop, as a function of the
feedback gain. This has a peak at almost the same value of feedback
gain as the kinetic energy has a minimum, as one would intuitively
expect for broadband excitation since the mechanism of vibration
control here is local damping.
[0035] The variation of absorbed power with gain suggests that this
may be a convenient way to self-tune the feedback gain, using only
local parameters to the controller, to achieve a minimum in the
kinetic energy, which is a global measure of performance. What is
more, the force applied by the controller in this case is, by
definition, equal to .gamma.v, where .gamma. is the feedback gain,
with units of Nsm.sup.-1 and v is the local upwards velocity so
that the averaged power absorbed, W, is equal to
W= fv=.gamma. v.sup.2,
where the overbar denotes time averaging. The measured velocity is
deliberately defined to be in the opposite direction to the applied
force so that .gamma. is a positive quantity for negative feedback.
The power absorbed can thus be estimated directly from the mean
square value of the measured velocity and the known feedback
gain.
[0036] FIG. 4 shows a block diagram of such a self-tuning vibration
controller apparatus comprising a controller 2 an actuator 1, and a
velocity sensor 4. The actuator 1 is attached to a panel 6. In the
illustrated arrangement, the instantaneous value of the measured
velocity is directly fed back to the ideal force actuator via the
gain .gamma., whose value is adjusted by an algorithm that
maximises the power absorbed, as estimated by .gamma. times the
mean square value of the measured velocity. For broadband
excitation, the power absorption curve in FIG. 3(b) has a unique
global maximum and so a number of algorithms could be used to
adjust .gamma. to maximise .gamma. v.sup.2.
[0037] Although the principle of self-tuning to maximise power
absorption can be readily demonstrated using idealised force
actuators, it is often not possible to use these in practice, since
there may be no solid structure to react the force against.
Inertial actuators react to the generated force off a proof mass
and have been widely used for active vibration control. Above their
natural frequency they can behave very much like ideal force
actuators over a frequency band of several decades, before higher
order resonances interfere with their dynamics. FIG. 5 shows the
blocked frequency response of a single degree of freedom model of
such a current-driven inertial actuator, with the parameters listed
in the table of FIG. 4, which has a natural frequency of about 10
Hz and a damping ratio of about 0.7. The response of the actuator
with .+-.20% variations in both its stiffness and damping are also
shown, for later use.
[0038] There are a number of additional problems encountered when
designing a self-tuning method for a velocity feedback loop with an
inertial actuator, compared with that using an ideal force
actuator. First, the feedback control loop is no longer
unconditionally stable, even under ideal conditions, since the
180.degree. phase shift in the response of the actuator below its
natural frequency will give rise to low frequency instabilities if
the feedback gain is too high, although an improvement in the
maximum gain can be achieved if a compensator is used. It is thus
important to adjust the feedback gain much more carefully than in
the case of an ideal force actuator, to avoid the system becoming
unstable, and so avoid the possibility of damage and enhancement of
vibration.
[0039] FIG. 6 shows the kinetic energy on the panel referred to
above when a direct velocity feedback loop with gain .gamma. is
implemented using an inertial actuator modelled as a single degree
of freedom system with the characteristics listed in table of FIG.
2. FIG. 6 illustrates feedback gains of .gamma.=0 (solid-line),
.gamma.=7 (dashed line), .gamma.=25 (faint line) and .gamma.=51.4
(dashed line). The response of the panel is now more damped, even
when the feedback gain is zero, due to the passive loading of the
actuator, which acts primarily as a passive damper above its
natural frequency. As the feedback gain is increased, significant
attenuation is initially obtained at the first few panel
resonances, as in FIG. 1 above. At higher gains, however, as well
as the additional resonances due to pinning starting to appear,
there is also now significant enhancement of the vibration at the
natural frequency of the actuator, due to the positive feedback in
this frequency region caused by the phase response of the actuator.
The feedback gain in this case, in which the actuator is driven by
a current, has units of Asm.sup.-1, but since the assumed
transduction coefficient, .phi..sub.a, is 2.6 NA.sup.-1, it has a
similar numerical value to that used above.
[0040] The frequency-averaged kinetic energy of the plate and local
absorbed power is plotted as a function of feedback gain in FIGS.
7(a) and 7(b) for this case. In FIGS. 7(a) and 7(b), frequency
averaged kinetic energy of the panel (a) and power absorbed by the
controller (b) as function of feedback gain for a local velocity
feedback controller are shown driving an inertial actuator with a
natural frequency of 10 Hz (solid line). Also plotted is the
estimated power absorbed when the actuator model is incorrectly
identified; +20% .omega..sub.0+20% .zeta. (dashed line), +20%
.omega..sub.0-20% .zeta. (dotted line), -20% .omega..sub.0+20%
.zeta. (dash-dotted line), -20% .omega..sub.0-20% .zeta. (faint
line). These graphs are similar to those in FIG. 1 until the
critical gain is approached for which the system becomes unstable.
An exception at low control gains is that the kinetic energy,
normalised by that before the actuator is attached is reduced and
the power absorbed by the controller no longer tends to zero. This
is because the passive response of the inertial actuator still
dissipates power even when the actuator is undriven. As the
feedback gain is increased towards the value for which the system
becomes unstable, however, the kinetic energy becomes very large
and the power absorbed becomes negative. For the arrangement
assumed here, the system is only stable for feedback gains below
about 52.
[0041] The frequency domain results are not valid for higher
feedback gains. It is striking how quickly these curves deviate
from those using an ideal force actuator as the instability is
approached, and it is as if the power absorbed falls off a
cliff.
[0042] Reference is now made to FIG. 8 which shows a vibration
control apparatus comprising an inertial actuator 10, a controller
12, and a velocity sensor 14. The actuator is attached to a panel
20. The controller 12 comprises a processor and an associated
memory to store machine readable instructions to be executed by the
processor.
[0043] The force supplied by the actuator 10 is also no longer
directly proportional to the input signal, since the actuator has
its own dynamics. These exhibit themselves in two ways, that can be
made clear using a superposition approach, assuming only that the
actuator is linear, so that the force supplied by the internal
actuator 10 to the structure 20 can be written as
f=T.sub.au+Z.sub.av
where we define
T a = f u | v = 0 and Z a = f v | u = 0 , ##EQU00001##
so that T.sub.a is the blocked frequency response of the actuator,
u is the input signal, which may be either voltage or current,
Z.sub.a is the undriven mechanical impedance of the actuator and v
is the local upward velocity.
[0044] In order to calculate the local power absorbed by the
actuator 10, as the product of the force it produces multiplied by
the local velocity, it is thus necessary to calculate an estimate
of the force, {circumflex over (f)}, using estimates of the blocked
response and undriven impedance {circumflex over (T)}.sub.a and
{circumflex over (Z)}.sub.a, so that
{circumflex over (f)}={circumflex over (T)}.sub.au+{circumflex over
(Z)}.sub.av
as illustrated in FIG. 8. FIG. 8 also shows how this estimate of
the absorbed power, {circumflex over ( fv, is used to tune the
feedback gain .gamma.. A compensator, C, is also included before
the actuator 10, which is assumed to be unity here, but in general
could be used to lower the apparent natural frequency of the
actuator, in which case T.sub.a and Z.sub.a would need to be
estimated with this compensator in place. It will be evident from
the above that the estimate of force, f is derived from gain
control signal, u and the measured velocity. It will be appreciated
that T.sub.a and Z.sub.a could be obtained for a generic type of
actuator, rather than from measurements on a specific case.
[0045] One of the potential dangers in this approach is that the
actuator dynamics are never known perfectly, and may change with
time or operating temperature. A series of further simulations have
thus been conducted with .+-.20% deviations in either the modelled
natural frequency or modelled damping ratio of the actuator, which
give rise to the modified actuator responses shown in FIG. 5. The
effect of these deviations in the modelled response on the
estimated power are also plotted in FIG. 7(b), which shows that
although the estimated power is somewhat in error for low feedback
gains, it retains the same shape as that with an accurate estimate
of applied force near its peak and can thus still be reliably used
to tune the feedback gain. When the feedback gain is very close to
instability, however, and the estimated natural frequency of the
actuator is below the true value, there is a sharp spike in the
estimated absorbed power. The true force is then very close to
being out of phase with the input signal, u, but the estimated
force will have less phase shift, since the phase of estimated
actuator response is lower than the true value, as can be seen in
FIG. 5. In FIG. 5, the blocked frequency response of an inertial
actuator, modelled as a single degree of freedom system with the
parameters shown in the table of FIG. 2 (solid line) and with
.+-.20% variations in its natural frequency and damping. +20%
.omega..sub.0+20% .zeta. (dashed line), +20% .omega..sub.0-20%
.zeta. (dotted line), -20% .omega..sub.0+20% .zeta. (dash-dotted
line), -20% .omega..sub.0-20% .zeta. (faint line). The estimated
absorbed power thus becomes greater than the true power, since the
large force and input signal appear to be closer to being in phase.
This effect should not prevent the convergence of a practical
controller, however, since it occurs so close to the point of
instability, which the controller must in any case steer clear of
at all cost.
[0046] The adaptation algorithm used to adjust the feedback gain
based on the estimated power absorbed would thus have to be
carefully designed not to stray too close to the unstable region.
This is particularly important if the inertial actuator did not
have such a low natural frequency, compared with the first
structural resonance, as that assumed above. In that case, the
maximum in the power absorption curve with an ideal force actuator
could occur at a significantly higher feedback gain than the
stability limit, so that the optimal feedback gain with the
inertial actuator is very close to the limit of stability. This is
illustrated in FIG. 9, in which the actuator stiffness is increased
so that its natural frequency is changed from 10 Hz to 20 Hz and
its damping ratio from 0.7 to 0.35. In FIG. 9, frequency averaged
kinetic energy of the panel (a), and power absorbed by the
controller (b) as a function of feedback gain for a local velocity
feedback controller driving an inertial actuator with a natural
frequency of 20 Hz. Also plotted is the estimated power absorbed
when the actuator model is incorrectly identified; +20%
.omega..sub.0+20% .omega. (dashed line), +20% .omega..sub.0-20%
.zeta. (dotted line), -20% .omega..sub.0+20% .zeta. (dash-dotted
line), -20% .omega..sub.0-20% .zeta. (faint line).
[0047] The ratio of the maximum, stable feedback gain,
.gamma..sub.max, to the optimum feedback gain, .gamma..sub.opt, can
be estimated by using the expression for these quantities which
are
.gamma. max .apprxeq. 2 .zeta. a M 1 .omega. 1 2 .omega. a
##EQU00002## .gamma. opt .apprxeq. 2 M .omega. 1 .pi.
##EQU00002.2##
where M is the mass of the panel, .omega..sub.l its first natural
frequency, M.sub.1 the model mass at this frequency, assumed to be
approximately M/JI, and .omega..sub.a and .zeta..sub.a are the
natural frequency and damping ratio of the actuator, so that
.gamma. max .gamma. opt .apprxeq. .zeta. a .omega. 1 .omega. a .
##EQU00003##
[0048] This ratio is greater than unity in the simulations
presented here when the actuator natural frequency is 10 Hz, as in
FIG. 7, but less than unity when the actuator natural frequency is
20 Hz, as in FIG. 9.
[0049] It will be appreciated that the measure of force referred to
above used to calculate the power absorbed, could be derived from
signals other than the gain control signal, u. For example, a
modified embodiment of the vibration control apparatus of FIG. 8
may include a force sensor to directly measure force, and the
output of the sensor received and processed by the control
arrangement.
[0050] A method and apparatus of automatically tuning the gain of a
local velocity feedback controller has been discussed, based on the
maximisation of the local absorbed power. Advantageously, it is
shown that for broadband excitation the feedback gain that
maximises the power absorbed by a local controller on a panel is
almost the same as that which minimises the panel's overall kinetic
energy.
[0051] In the case of an inertial actuator the applied force is
inferred from the measured velocity, control signal and the
modelled response and input impedance of the actuator. The
estimated power absorbed by the inertial actuator is a good
approximation to its true value even if there are significant
differences between the true values of the actuator's natural
frequency and damping ratio and the estimated values. This
demonstrates that this approach to self-tuning is robust to the
kind of changes in the response of the actuator that are likely to
occur over time or with changing temperature. If the actuators are
constructed to a reasonable tolerance, it may be possible to use a
single model of their response in all manufactured feedback control
units.
[0052] One aspect of self-tuning with the use of inertial actuators
is the need to avoid feedback gains for which the system becomes
unstable, since this will cause significant enhancement of the
vibration and, potentially, damage. The optimal feedback gain can
be kept well below the unstable limit provided the actuator
resonance frequency is well below the first natural frequency of
the panel and the actuator is well damped, although this is not
always possible in practice. The maximum stable feedback gain also
depends on the dynamics of the structure to which the controller is
attached and on the number of local control units on the structure.
It may thus be necessary in these cases to develop supplementary
methods of assessing how close the feedback gain is to the unstable
limit, so that this can be avoided. It will be appreciated that the
control problem becomes significantly harder if the actuators are
not well suited to feedback control on the structure being
controlled.
* * * * *